This completes the proof of Claim (2): almost surely eventually, there exist roots \(\tilde{\lambda }_{(1)}, \ldots , \tilde{\lambda }_{(i_0)}\)  respectively encircled by \(\gamma _{1,n}, \ldots , \gamma _{i_0, n}\) . Claim (3)—that \(\tilde{\lambda }_{(1)}, \ldots , \tilde{\lambda }_{(i_0)}\)  are eigenvalues of \(\widetilde{S}_n\) —follows from Lemma REF  and the fact that \(\widetilde{M}_n(z)\)  is invertible for \(z \notin \mathbb {R}\)  (Lemma A.1 of [1]}).
